a) Prove that Lipschitz functions are continuous.
b) Prove that |x| is Lip([-1,1]), i.e. Lipschitz functions can
have corners.
Extra problem 1.2 (section 1.3):
We proved in class that W^1_p (I) is "contained" in C^0 (I)
for n=1, I=[a,b].
a) Prove that the statement is not true if I is not bounded by
finding a counter-example.
b) Containment the other way is in general not true, however
for p=1 it is true for absolutely continuous functions - see exercise 17
(look at, do not prove).
Extra problem 1.3 (section 1.4):
Suppose u is in W^1_p (R) 1<= p < infty. Prove that
lim(as x goes to +- infty) u(x) =0.
(Hint: Use the fact that C^{infty}_0 is dense in W^1_0 and the
continuous imbedding of W^1_p in L^{infty}). Is this
true for p=infinity? Why?
Extra problem 1.4 (section 1.4):
Let u,v lie in W^1_p (I), I a bounded
domain in 1-dimension, 1 <= p < infty. Prove that
uv is in W^1_p (I), and (uv)'=u'v+uv'
(Hint: Use same hint as in Extra problem 1.3).
This is also true for p= infty, but must be proven a different way
- try this as well (hint: use W^1_{infty} is contained in W^1_1(I)
).
Extra problem 1.4 (section
1.4):
Show that |||u|||_{W^1_p} = ||u||_p + sum ||\frac{partial u}{partial x_i}||_p
and ||u|| _{W^1_p} = ||u||_p^p + sum ||\frac{partial u}{partial x_i}||_p^p]^{1/p}
are equivalent norms
Ch 2:
1, 5, 7, 10, 11
Extra problem 2.1 (section
2.9):
Prove that the problem of section 2.9 is continuous and coercive.
Ch 3:
3, 9, 13, 26 (specific example
will suffice), 30
Extra Problem 3.1
(section 3.4):
As in class find the basis functions in terms of the barycentric
coordinates for a tetrahedron in R^3 for k=1 and k=2
Extra Problem 3.2 (section
3.5):
Determine the basis function for the serendipity element on the rectangle
with
vertices (1,1), (-1,1), (-1,-1), (1,-1). Which terms of
c_0 + c_1 x + c_2 y + c_3 x^2 + c_4 xy + c_5 y^2 + c_6 xy^2 + c_7 x^2 y
+ c_7 x^2 y
+ c_8 x^2 y^2
are missing?